\usepackage{optprog}
\usepackage[dvips]{epsfig}
\usepackage[dvips]{graphicx}
\usepackage{color}
\usepackage{soul}
\usepackage{setspace}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{bbold}
\usepackage{caption}
\usepackage{lineno}
\usepackage[strict]{changepage}
\usepackage{MnSymbol,marvosym}
\usepackage[ruled]{algorithm}
\usepackage{lscape,framed}
\usepackage{rotating}
\usepackage{tikz,xifthen}
\usepackage[absolute,overlay]{textpos}
\usepackage{algpseudocode}
\usepackage{geometry}
\usepackage{verbatim} %for multiline comments with \begin{comment}
\usepackage{ctable,multirow,subfigure,enumitem}
\usepackage[pdftex]{hyperref}
\parindent=0pt
\parskip=10pt
\hyphenpenalty = 10000
\hypersetup{
colorlinks,%
citecolor=black,%
filecolor=black,%
linkcolor=black,%
urlcolor=black
}

%the next set of commands are for representing the different number sets
% for example, instead of writing \mathbb{R}, \R will do
\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}
\DeclareMathSymbol{\N}{\mathbin}{AMSb}{"4E}
\DeclareMathSymbol{\Z}{\mathbin}{AMSb}{"5A}
\DeclareMathSymbol{\R}{\mathbin}{AMSb}{"52}
\DeclareMathSymbol{\Q}{\mathbin}{AMSb}{"51}
\DeclareMathSymbol{\I}{\mathbin}{AMSb}{"49}
\DeclareMathSymbol{\C}{\mathbin}{AMSb}{"43}
\DeclareMathOperator{\conv}{conv}

\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{claim}[thm]{Claim}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{obs}[thm]{Observation}
\newtheorem{conj}{Conjecture}

\theoremstyle{definition}
\newtheorem{dfn}{Definition}
\newtheorem*{example}{Example}
\newtheorem{problem}{Open Problem}
\newtheorem{rem}{Remark}
\newtheorem{assumption}{Assumption}

%add desired commands here
\newcommand\ignore[1]{}
\newcommand{\RR}{\ensuremath{\mathbb{R}}}
\newcommand\Rnn{\R_{\ge0}}
\newcommand\Rsym[1]{\R_\mathsf{sym}^{#1 \times #1}}
\newcommand\Snn{\mathcal{S}_{\ge 0}}
\newcommand\set[2]{\left\{ {#1} \ : \ {#2} \right\}}
\newcommand{\abs}[1]{|#1|} % for absolute value
\newcommand{\norm}[1]{\parallel#1\parallel} % for the norm of a vector
\newcommand\ceil[1]{\left\lceil{#1}\right\rceil}
\newcommand\floor[1]{\left\lfloor{#1}\right\rfloor}
\newcommand\x{\mathbf{x}}
\newcommand\y{\mathbf{y}}
\newcommand\Y{\mathbf{Y}}
\newcommand{\K}{\mathcal K}
\newcommand{\F}{\ensuremath{{\mathcal F}}}
\newcommand{\REV}[1]{#1}
\newcommand\z{\mathbf{z}}
\newcommand{\lc}{\preccurlyeq}
\newcommand\p{\mathbf{p}}
\renewcommand\P{\mathcal{P}}
\newcommand\q{\mathbf{q}}
\newcommand{\stcomp}[1]{{#1}^{\mathsf{C}}}
\newcommand{\lprelax}{\operatorname{relax}}
\renewcommand\r{\mathbf{r}}
\renewcommand\a{\mathbf{a}}
\renewcommand\b{\mathbf{b}}
\renewcommand\v{\mathbf{v}}
\newcommand\id{\mathrm{Id}}
\newcommand\cone{\mathrm{cone}}
\newcommand\define{\,:=\,}
\newcommand\inset{\,\in\,}
\renewcommand\implies{\hspace{3mm}\Rightarrow\hspace{3mm}}
\newcommand\relint{\mathrm{relint}}
\renewcommand\setminus{\,\backslash\,}
\newcommand\bs[1]{\boldsymbol{#1}}
\newcommand{\HRule}{\rule{\linewidth}{0.5mm}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\renewcommand{\refname}{Bibliography}
\newcommand\binset{\{0,1\}}
\newcommand\sm{\setminus}
\newcommand\update{\leftarrow}
\newcommand{\solver}[1]{$\text{\texttt{#1}}^\copyright\,$}
\newcommand{\benn}{\begin{equation}\nonumber}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bpi}{\boldsymbol\pi}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\atam}{Atamt\"{u}rk }
\newcommand{\tbc}{\textit{To be continued...}}
\newcommand{\tba}{\textit{To be added...}}
\newcommand{\revise}[1]{\textcolor{red}{#1}}
\renewcommand{\refname}{References}

\topmargin=-0.5in      %top margin is 1/2 inch (note negative value)
\oddsidemargin=0.1in    %left margin is 1 inch on right-hand pages
\evensidemargin=0.1in   %same for left-hand pages in two sided
\textwidth=6.5in      %leaves 1 inch for the right margin
\textheight=9in       %9 inches reserved for the text.


\def\BCOLHeader {
\begin{center}
\vspace*{-2.5cm}
\begin{tikzpicture}%
             \node [opacity=0.1]{\includegraphics[width=5.38cm]{ucbseal_line_k}};%
  \end{tikzpicture}
\vspace*{1cm}
\end{center}
\begin{textblock}{8}(4,2.3)
\begin{center}
{\bfseries \Large BCOL RESEARCH REPORT}\\[3mm]
Industrial Engineering \& Operations Research\\
University of California, Berkeley, CA, $94709-1777$
\end{center}
\end{textblock}
}

\newcommand{\funding}[1][\@empty]{%
\ifx\@empty#1  {}  \else  {\footnotetext{This research has been supported, in part, by #1}}  \fi
}

\newcommand\affiliation[2] {
\footnotetext{#1:#2}
}

\newcommand{\key}[1]{{\scriptsize\textsl{Keywords} : #1}}
